Abstract:
The survey reviews recent results on spectral analysis of differential and finite-difference operators with random spatially homogeneous coefficients. The corresponding problems that crystallized in the development of a number of areas in mathematics and related sciences are very rich and diverse. We discuss the traditional problems of spectral analysis, where the use of probabilistic ideas and methods now allows highly detailed spectral analysis to be performed for an essentially broader class of operators, as well as new problems and results obtained in the framework of this theory.
Citation:
L. A. Pastur, “Spectral theory of random self-adjoint operators”, Itogi Nauki i Tekhniki. Ser. Teor. Veroyatn. Mat. Stat. Teor. Kibern., 25, VINITI, Moscow, 1987, 3–67; J. Soviet Math., 46:4 (1989), 1979–2021
\Bibitem{Pas87}
\by L.~A.~Pastur
\paper Spectral theory of random self-adjoint operators
\serial Itogi Nauki i Tekhniki. Ser. Teor. Veroyatn. Mat. Stat. Teor. Kibern.
\yr 1987
\vol 25
\pages 3--67
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/intv67}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=927296}
\zmath{https://zbmath.org/?q=an:0661.60079|0684.60051}
\transl
\jour J. Soviet Math.
\yr 1989
\vol 46
\issue 4
\pages 1979--2021
\crossref{https://doi.org/10.1007/BF01096021}
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https://www.mathnet.ru/eng/intv67
https://www.mathnet.ru/eng/intv/v25/p3
This publication is cited in the following 5 articles:
Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups”, Proc. Steklov Inst. Math., 306 (2019), 196–211
V. Katsnelson, Interpolation Theory, Systems Theory and Related Topics, 2002, 243
I. A. Koshovets, “Unitary analog of the Anderson model. Purely point spectrum”, Theoret. and Math. Phys., 89:3 (1991), 1249–1270
W. Kirsсh, S. A. Molchanov, L. A. Pastur, “One-dimensional Schrödinger operator with unbounded potential: The pure point spectrum”, Funct. Anal. Appl., 24:3 (1990), 176–186
L. V. Bogachev, S. A. Molchanov, “Mean-field models in the theory of random media. I”, Theoret. and Math. Phys., 81:2 (1989), 1207–1214
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